Recent Trends in Signal Representations and Their Role in Image Processing Michael Elad The CS Department The Technion – Israel Institute of technology Haifa 32000, Israel Research Day on Image Analysis and Processing Industrial Affiliate Program (IAP) The Computer Science Department – The Technion January 27 th, 2005
Recent Advances in Signal Representations and Applications to Image Processing 2 Today’s Talk is About Sparsity and Overcompleteness We will show today that Sparsity & Overcompleteness can & should be used to design new powerful signal/image processing tools (e.g., transforms, priors, models, …), The obtained machinery works very well – we will show these ideas deployed to applications.
Recent Advances in Signal Representations and Applications to Image Processing 3 Agenda 1.A Visit to Sparseland Motivating Sparsity & Overcompleteness 2.Problem 1: Transforms & Regularizations How & why should this work? 3.Problem 2: What About D? The quest for the origin of signals 4.Problem 3: Applications Image filling in, denoising, separation, compression, … Welcome to Sparseland
Recent Advances in Signal Representations and Applications to Image Processing 4 Generating Signals in Sparseland MM K N A fixed Dictionary Every column in D (dictionary) is a prototype signal (Atom). The vector is generated randomly with few non-zeros in random locations and random values. A sparse & random vector N
Recent Advances in Signal Representations and Applications to Image Processing 5 Simple: Every generated signal is built as a linear combination of few atoms from our dictionary D Rich: A general model: the obtained signals are a special type mixture- of-Gaussians (or Laplacians). Sparseland Signals Are Special Multiply by D MM
Recent Advances in Signal Representations and Applications to Image Processing 6 Transforms in Sparseland ? MM MM Assume that x is known to emerge from.. We desire simplicity, independence, and expressiveness. MM How about “Given x, find the α that generated it in ” ? TT
Recent Advances in Signal Representations and Applications to Image Processing 7 Known So, In Order to Transform … We need to solve an under-determined linear system of equations: Sparsity is measured using the L 0 norm: Among all (infinitely many) possible solutions we want the sparsest !!
Recent Advances in Signal Representations and Applications to Image Processing 8 Are there practical ways to get ? Signal’s Transform in Sparseland Multiply by D A sparse & random vector 4 Major Questions Is ? Under which conditions? How effective are those ways? How would we get D?
Recent Advances in Signal Representations and Applications to Image Processing 9 Inverse Problems in Sparseland ? Assume that x is known to emerge from. M MM Suppose we observe, a “blurred” and noisy version of x with. How will we recover x? Noise How about “find the α that generated the x …” again? Q
Recent Advances in Signal Representations and Applications to Image Processing 10 Inverse Problems in Sparseland ? Multiply by D A sparse & random vector “blur” by H 4 Major Questions (again!) How would we get D? Are there practical ways to get ? How effective are those ways? Is ? How far can it go?
Recent Advances in Signal Representations and Applications to Image Processing 11 Back Home … Any Lessons? Several recent trends worth looking at: JPEG to JPEG From (L 2 -norm) KLT to wavelet and non- linear approximation From Wiener to robust restoration – From L 2 -norm (Fourier) to L 1. (e.g., TV, Beltrami, wavelet shrinkage …) From unitary to richer representations – Frames, shift-invariance, bilateral, steerable, curvelet Approximation theory – Non-linear approximation ICA and related models Sparsity. Overcompleteness. Sparsity. Sparsity & Overcompleteness. Independence and Sparsity. Sparseland is HERE
Recent Advances in Signal Representations and Applications to Image Processing 12 To Summarize so far … We do! this model is relevant to us just as well. Who cares? So, what are the implications? Sparsity? looks complicated (a) Practical solvers? (b) How will we get D? (c) Applications? The Sparseland model for signals is interesting since it is rich and yet every signal has a simple description MM We need to solve (or approximate the solution of) a linear system with more unknowns than equations, while finding the sparsest possible solution
Recent Advances in Signal Representations and Applications to Image Processing 13 Agenda 1.A Visit to Sparseland Motivating Sparsity & Overcompleteness 2.Problem 1: Transforms & Regularizations How & why should this work? 3.Problem 2: What About D? The quest for the origin of signals 4.Problem 3: Applications Image filling in, denoising, separation, compression, … T Q
Recent Advances in Signal Representations and Applications to Image Processing 14 Lets Start with the Transform … known Put formally, Our dream for Now: Find the sparsest solution of Known
Recent Advances in Signal Representations and Applications to Image Processing 15 Questions to Address Is ? Under which conditions? Are there practical ways to get ? How effective are those ways? How would we get D? 4 Major Questions
Recent Advances in Signal Representations and Applications to Image Processing 16 Multiply by D MM Question 1 – Uniqueness? Suppose we can solve this exactly Why should we necessarily get ? It might happen that eventually.
Recent Advances in Signal Representations and Applications to Image Processing 17 Matrix “Spark” Rank = 4 Spark = 3 Example: Donoho & Elad (‘02) Definition: Given a matrix D, =Spark{D} is the smallest and and number of columns that are linearly dependent.
Recent Advances in Signal Representations and Applications to Image Processing 18 Uniqueness Rule Suppose this problem has been solved somehow This result implies that if generates signals using “sparse enough” , the solution of the above will find it exactly. MM If we found a representation that satisfy Then necessarily it is unique (the sparsest). Uniqueness Donoho & Elad (‘02)
Recent Advances in Signal Representations and Applications to Image Processing 19 Are there reasonable ways to find ? MM Question 2 – Practical P 0 Solver? Multiply by D
Recent Advances in Signal Representations and Applications to Image Processing 20 Matching Pursuit (MP) Mallat & Zhang (1993) Next steps: given the previously found atoms, find the next one to best fit … The Orthogonal MP (OMP) is an improved version that re-evaluates the coefficients after each round. The MP is a greedy algorithm that finds one atom at a time. Step 1: find the one atom that best matches the signal.
Recent Advances in Signal Representations and Applications to Image Processing 21 Basis Pursuit (BP) Instead of solving Solve Instead Chen, Donoho, & Saunders (1995) The newly defined problem is convex. It has a Linear Programming structure. Very efficient solvers can be deployed: Interior point methods [Chen, Donoho, & Saunders (`95)], Sequential shrinkage for union of ortho-bases [Bruce et.al. (`98)], If computing Dx and D T are fast, based on shrinkage [Elad (`05)].
Recent Advances in Signal Representations and Applications to Image Processing 22 Multiply by D MM Question 3 – Approx. Quality? How effective are the MP/BP in finding ?
Recent Advances in Signal Representations and Applications to Image Processing 23 Evaluating the “Spark” The Mutual Incoherence is a property of the dictionary (just like the “Spark”). The smaller it is, the better the dictionary. The Mutual Incoherence M is the largest entry in absolute value outside the main diagonal of D T D. DTDT = D DTDDTD Compute Assume normalized columns
Recent Advances in Signal Representations and Applications to Image Processing 24 BP and MP Equivalence Given a signal x with a representation, Assuming that, BP and MP are Guaranteed to find the sparsest solution. Donoho & Elad (‘02) Gribonval & Nielsen (‘03) Tropp (‘03) Temlyakov (‘03) Equivalence MP is typically inferior to BP! The above result corresponds to the worst-case. Average performance results are available too, showing much better bounds [Donoho (`04), Candes et.al. (`04), Elad and Zibulevsky (`04)].
Recent Advances in Signal Representations and Applications to Image Processing 25 What About Inverse Problems? “blur” by H We had similar questions regarding uniqueness, practical solvers, and their efficiency. It turns out that similar answers are applicable here due to several recent works [Donoho, Elad, and Temlyakov (`04), Tropp (`04), Fuchs (`04)]. Multiply by D A sparse & random vector
Recent Advances in Signal Representations and Applications to Image Processing 26 To Summarize so far … The Sparseland model for signals is relevant to us. We can design transforms and priors based on it Use pursuit Algorithms Find the sparsest solution? A sequence of works during the past 3-4 years gives theoretic justifications for these tools behavior Why works so well? What next? (a)How shall we find D? (b)Will this work for applications? Which?
Recent Advances in Signal Representations and Applications to Image Processing 27 Agenda 1.A Visit to Sparseland Motivating Sparsity & Overcompleteness 2.Problem 1: Transforms & Regularizations How & why should this work? 3.Problem 2: What About D ? The quest for the origin of signals 4.Problem 3: Applications Image filling in, denoising, separation, compression, …
Recent Advances in Signal Representations and Applications to Image Processing 28 Problem Setting Multiply by D MM Given these P examples and a fixed size [N K] dictionary D: 1. Is D unique? 2. How would we find D?
Recent Advances in Signal Representations and Applications to Image Processing 29 Uniqueness? If is rich enough* and if then D is unique. Uniqueness Aharon, Elad, & Bruckstein (`05) Comments: MM “Rich Enough”: The signals from could be clustered to groups that share the same support. At least L+1 examples per each are needed. This result is proved constructively, but the number of examples needed to pull this off is huge – we will show a far better method next. A parallel result that takes into account noise can be constructed similarly.
Recent Advances in Signal Representations and Applications to Image Processing 30 Each example has a sparse representation with no more than L atoms Each example is a linear combination of atoms from D Practical Approach – Objective D X A
Recent Advances in Signal Representations and Applications to Image Processing 31 The K–SVD Algorithm – General D Initialize D Sparse Coding Use MP or BP Dictionary Update Column-by-Column by SVD computation Aharon, Elad, & Bruckstein (`04) X
Recent Advances in Signal Representations and Applications to Image Processing 32 K–SVD Sparse Coding Stage D X For the j th example we solve Pursuit Problem !!!
Recent Advances in Signal Representations and Applications to Image Processing 33 K–SVD Dictionary Update Stage D G k : The examples in and that use the column d k. The content of d k influences only the examples in G k. Let us fix all A apart from the k th column and seek both d k and the k th column to better fit the residual!
Recent Advances in Signal Representations and Applications to Image Processing 34 d k is obtained by SVD on the examples’ residual in G k. K–SVD Dictionary Update Stage D Residual E We should solve:
Recent Advances in Signal Representations and Applications to Image Processing 35 To Summarize so far … The Sparseland model for signals is relevant to us. In order to use it effectively we need to know D Use the K-SVD algorithm How D can be found? We have established a uniqueness result and shown how to practically train D using the K-SVD Will it work well? What next? (a)Deploy to applications (b)Generalize in various ways: multiscale, non-negative factorization, speed-up, … (c)Performance guarantees?
Recent Advances in Signal Representations and Applications to Image Processing 36 Agenda 1.A Visit to Sparseland Motivating Sparsity & Overcompleteness 2.Problem 1: Transforms & Regularizations How & why should this work? 3.Problem 2: What About D? The quest for the origin of signals 4.Problem 3: Applications Image filling in, denoising, separation, compression, …
Recent Advances in Signal Representations and Applications to Image Processing 37 Inpainting (1) [Elad, Starck, & Donoho (`04)] Source Outcome predetermined dictionary: Curvelet+DCT
Recent Advances in Signal Representations and Applications to Image Processing 38 Inpainting (2) Source Outcome predetermined dictionary: Curvelet+DCT
Recent Advances in Signal Representations and Applications to Image Processing 39 10,000 sample 8-by-8 images. 441 dictionary elements. Approximation method: MP. K-SVD: Overcomplete Haar K-SVD on Images [Aharon, Elad, & Bruckstein (`04)]
Recent Advances in Signal Representations and Applications to Image Processing 40 Filling-In Missing Pixels 90% missing pixels Haar Results Average # coefficients 4.48 RMSE: K-SVD Results Average # coefficients 4.27 RMSE: 25.32
Recent Advances in Signal Representations and Applications to Image Processing 41 Filling-In Missing Pixels
Recent Advances in Signal Representations and Applications to Image Processing 42 K-SVD dictionary Haar dictionary DCT dictionary OMP with error bound Compression
Recent Advances in Signal Representations and Applications to Image Processing 43 DCT Results BPP = 1.01 RMSE = 8.14 K-SVD Dictionary BPP = RMSE = 7.67 Haar Dictionary BPP = RMSE = 8.41 Compression
Recent Advances in Signal Representations and Applications to Image Processing 44 Compression
Recent Advances in Signal Representations and Applications to Image Processing 45 Today We Have Discussed 1.A Visit to Sparseland Motivating Sparsity & Overcompleteness 2.Problem 1: Transforms & Regularizations How & why should this work? 3.Problem 2: What About D ? The quest for the origin of signals 4.Problem 3: Applications Image filling in, denoising, separation, compression, …
Recent Advances in Signal Representations and Applications to Image Processing 46 Summary Sparsity and Over- completeness are important ideas that can be used in designing better tools in signal/image processing We are working on resolving those difficulties: Performance of pursuit alg. Speedup of those methods, Training the dictionary, Demonstrating applications, … There are difficulties in using them! Future transforms and regularizations will be data- driven, non-linear, overcomplete, and promoting sparsity. The dream?